2. Mean syzygy calculation
The Chinese had their own traditional method to record dates, named Ganzhi, the same as Shoushi Calendar. In ancient China, the period of "Ganzhi" (i.e. the Heavenly Stems and Earthly Branches) is 60 days (from 0 to 59 in this paper). It is an independent method from very ancient times, day by day, one period by one period. Each day in one period has its own name, the name is also called one Ganzhi. Generally the data (Ganzhi number) is integer, but in this paper, due to our calculation: INT(Ganzhi) is as the date number, its decimal part (Ganzhi - INT(Ganzhi)) is the time (Beijing's apparent solar time, long. E, lat. N), and the unit is the day. So Ganzhi is also a kind of date recording method.
According to the Shoushi Calendar, the Ganzhi of mean new Moon, , just before the Winter Solstice (AD 1280) is
with QY, named Qiying, days from the epoch to the first day of the period of Ganzhi which includes it, QY=55.0600. The "epoch" is the time of Winter Solstice in AD 1280. The number of Ganzhi is 55, the decimal part is the time, beginning at midnight, the unit is the day; RY, named Runying, days from the epoch to the new Moon which is just before it, RY=20.2050. The earlier adopted value is 20.1850 in the Shoushi Calendar. The name of Ganzhi is Wuxu (the number of Ganzhi is 34, the Gregorian date is 23 Dec. 1280), the time is 20:31 (Beijing's apparent solar time). This result is due to the real measurement of that time. The time of other mean new Moons equals to this value plus several lunar months B (named Shuoshi, B= ). The theory is very convenient, but the calculation is very complicated.
Generally, the Ganzhi of the Winter Solstice of the year before, , is given by
In this formula, the function MOD(X, Y) mod(X, Y), when mod(X, Y) 0; MOD(X, Y) (mod(X, Y)+Y), when mod(X, Y) 0; N year-1281 (named Jusuan or Jilian); , (named Suishi, days of tropical year, A is the mean value, A= ); , (named Xunzhou or Jifa, period of Ganzhi).
So (days from the Winter Solstice of the year before to the mean new Moon just before it) is
The Ganzhi of every mean new Moon in a given year could be calculated as
with n, the numbers of the syzygys from the mean syzygy of just before the last Winter Solstice (beginning at ).
The Ganzhi of every mean solar term in the year, , is given by
In these two equations, the result represents the number of Ganzhi (integral part) and its decimal, the unit is days, beginning at midnight. When , the represents the Ganzhi of the Winter Solstice just before the year, and the is the Ganzhi of the mean new Moon just before the Winter Solstice.
If the parameters N (year-1281) and n are given, according to Eq. (1) and (2), we could compute the Ganzhi of mean new Moon and that of mean solar term. This just is the method of the Shoushi Calendar to reckon mean syzygys and solar terms.
The key to this part is to get the beginning point of the mean syzygy and solar term of the year for calculations. The other mean values of this year are equal to these values in addition to several (n) B for syzygys or several (n) A' for solar terms.
© European Southern Observatory (ESO) 1998
Online publication: March 30, 1998